One-bit compressed sensing with partial Gaussian circulant matrices

Abstract

In this paper we consider memoryless one-bit compressed sensing with randomly subsampled Gaussian circulant matrices. We show that in a small sparsity regime and for small enough accuracy δ⁠, m≃δ−4slog(N/sδ) measurements suffice to reconstruct the direction of any s-sparse vector up to accuracy δ via an efficient program. We derive this result by proving that partial Gaussian circulant matrices satisfy an ℓ1/ℓ2 restricted isometry property property. Under a slightly worse dependence on δ⁠, we establish stability with respect to approximate sparsity, as well as full vector recovery results, i.e., estimation of both vector norm and direction.